https://matjournals.net/engineering/index.php/JOSME/issue/feed 2026-09-19T04:35:52+00:00 Open Journal Systems <p><strong>JOSME</strong> is a peer reviewed journal in the discipline of Applied Science published by the MAT Journals Pvt. Ltd. It is a print and e-journal focused towards the rapid publication of fundamental research papers on all areas of Statistics and Mathematical Engineering. Mathematical statistics is the application of mathematics to statistics, which was originally conceived as the science of the state the collection and analysis of facts about a country: its economy, land, military, population, and so forth.</p> https://matjournals.net/engineering/index.php/JOSME/article/view/4144 2026-09-19T04:35:52+00:00 V. Rangaveni venidtknl@gmail.com

<p><em>This paper examines a recurring difficulty in introductory algebra: the word inverse is used both for an element within a structure and for a map between structures. Although these ideas are related, they answer different mathematical questions. A transformation may be irreversible, while the monoid to which it belongs can still be isomorphic to another monoid. Similarly, every element of a group may have an inverse even when a homomorphism from that group is not one-to-one. Using a two-state transformation system, this theoretical commentary develops a connected set of examples covering semigroups, monoids, groups, subgroups, homomorphisms, and isomorphisms. The examples include reversible and irreversible transformations, a semigroup without identity, an isomorphism produced by relabeling states, modular reduction, and a bijection that does not preserve an operation. These contrasts clarify the distinctions among element invertibility, operation preservation, and bijectivity. The paper proposes a task-analysis framework that identifies the mathematical evidence required for each claim, such as an inverse, a homomorphism condition, or a counterexample. The aim is to support more precise teaching and assessment of algebraic structures in second-year engineering mathematics and introductory abstract algebra.</em></p>

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